Problem: Gustavo and Aiden are walking due west through the forest when they happen upon an angry grizzly bear. Gustavo runs away $23^\circ$ north of east for $300 \,\text{m}$, and Aiden runs away $33^\circ$ south of east for $250 \,\text{m}$. How far are Gustavo and Aiden away from each other? Do not round during your calculations. Round your final answer to the nearest meter.
Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $AB=d$. $56^\circ$ $d$ $250\text{ m}$ $300\text{ m}$ $A$ $B$ $C$ Since we are given two side lengths and the angle measure between them, we can use the law of cosines. Using the law of cosines $\begin{aligned} (AB)^2&=(AC)^2+(BC)^2-2AC\!\cdot\! BC\!\cdot\!\cos(C)\\\\ d^2&=300^2+250^2-2\cdot 300\cdot 250\cdot\cos(56^\circ) \gray{\text{Substitute}}\\\\ d&=\sqrt{300^2+250^2-2\cdot 300\cdot 250\cdot\cos(56^\circ)}\\\\ d&\approx 262 \end{aligned}$ The answer Gustavo and Aiden are $262$ meters from each other.